Control of pendulum oscillations with variable parameters

Two pendulum control problems are considered, in which oscillations are excited by changing the length (or the position of the center of mass) of the pendulum or by displacing the suspension point. The control objective is approaching a certain invariant set in the state space of the system. The solution approach is based on the speed gradient method. The obtained results make it possible to determine the domain of initial data and parameters on which the system possesses the desired properties.     

The stability of the equilibrium of a pendulum of variable length

The frequencies and modes of parametric oscillations of a pendulum of variable length for values of the modulation index from the smallest to the limit admissible values are investigated. The limits of the resonance zones of the first four oscillation modes are constructed and investigated by analytical and numerical methods, and the fundamental qualitative properties of the higher modes are established. Complete degeneracy of the modes with even numbers, i.e., coincidence of the frequencies of the odd and even eigenmodes for admissible values of the modulation parameter, is proved. The global pattern of the limits of the regions of stability of the lower position of equilibrium is constructed and it is shown that it differs considerably from the Ince–Strutt diagrams. Specific properties of the eigenmodes are established.     

The dynamics of a spherical pendulum with a vibrating suspension

The motion of a spherical pendulum whose point of suspension performs high-frequency vertical harmonic oscillations of small amplitude is investigated. It is shown that two types of motion of the pendulum exist when it performs high-frequency oscillations close to conical motions, for which the pendulum makes a constant angle with the vertical and rotates around it with constant angular velocity. For the motions of the first and second types the centre of gravity of the pendulum is situated below and above the point of suspension, respectively. A bifurcation curve is obtained, which divides the plane of the parameters of the problem into two regions. In one of these only the first type of motion can exist, while in the other, in addition to the first type of motion, there are two motions of the second type. The problem of the stability of these motion of the pendulum, close to conical, is solved. It is shown that the first type of motion is stable, while of the second type of motion, only the motion with the higher position of the centre of gravity is stable.     

The stability of an inverted pendulum with a vibrating suspension point

The problem of stabilizing the upper vertical (inverted) position of a pendulum using vibration of the suspension point is considered. The periodic function describing the vibrations of the suspension point is assumed to be arbitrary but possessing small amplitudes, and slight viscous damping is taken into account. A formula is obtained for the limit of the region of stability of the solutions of Hill's equation with damping in the neighbourhood of the zeroth natural frequency. The analytical and numerical results are compared and show good agreement. An asymptotic formula is derived for the critical stabilization frequency of the upper vertical position of the pendulum. It is shown that the effect of viscous damping on the critical frequency is of the third-order of smallness and, in all the examples considered, when viscous damping is taken into account the critical frequency increases.     

Trajectory tracking control of variable length pendulum by partial energy shaping

This paper concerns a trajectory tracking control problem for a pendulum with variable length, which is an underactuated mechanical system of two degrees-of-freedom with a single input of adjusting the length of the pendulum. We aim to study whether it is possible to design a time-invariant control law to pump appropriate energy into the variable length pendulum for achieving a desired swing motion (trajectory) with given desired energy and length of the pendulum. First, we show that it is difficult to avoid singular points in the controller designed by using the conventional energy-based control approach in which the total mechanical energy of the pendulum is controlled. Second, we present a tracking controller free of singular points by using only the kinetic energy of rotation and the potential energy of the pendulum and not using the kinetic energy of the motion along the rod. Third, we analyze globally the motion of the pendulum and clarify the stability issue of two closed-loop equilibrium points; and we also provide some conditions on control parameters for achieving the tracking objective. Finally, we show numerical simulation results to validate the presented theoretical results.     

A new first integral corresponding to Lyapunov's function for a pendulum of variable length

Summary Using Noether's theorem and the generalized Killing equations [1], new first integrals of the differential equation of motion for a class of non-conservative mechanical systems with one degree of freedom, a special case of which is a simple pendulum of variable length, are obtained. These integrals are identified as Lyapunov's functions for non-autonomous systems. The stability conditions are established.

---

Zusammenfassung Mit Hilfe des Noetherschen Satzes und den verallgemeinerten Killingschen Gleichungen werden neue erste Integrale der Bewegungsdifferentialgleichungen für eine Klasse von nichtkonservativen mechanischen Systemen mit einem Freiheitsgrad, die das Pendel mit veränderlicher Länge als Sonderfall enthält, hergeleitet. Diese Integrale stellen Ljapunovsche Funktionen dar, mit denen sich die Stabilitätsbedingungen ergeben.

     

A qualitative investigation of the oscillations of a pendulum with a periodically varying length and a mathematical model of a swing

The behaviour of the amplitude-frequency characteristics of families of periodic solutions, produced from the equilibrium position of a system, is established by a qualitative investigation of the equation of the oscillations of a pendulum, the length of which is an arbitrary periodic function of time. The non-local conditions for their stability and instability, expressed in terms of the amplitude and frequency of the oscillations, are obtained. The results are used when discussing the parametric and self-excited oscillatory model of a swing. In the parametric model the length of a swing is a specified periodic function of time, and in the self-excited oscillatory model it is a function of the phase coordinates of the system. For an appropriate choice of these functions, both systems have a common periodic solution. It is shown that the parametric model leads to an erroneous conclusion regarding the instability of the periodic mode, which is in fact realized in the oscillations of a swing, whereas the self-excited oscillatory model indicates its stability.     

Vibration reduction of a three DOF non-linear spring pendulum

The dynamic response of mechanical and civil structures subject to high-amplitude vibration is often dangerous and undesirable. Vibrations and dynamic chaos should be controlled or eliminated in both structures and machines. This can be employed via passive and active control methods. In this paper, a tuned absorber, in the transversally direction, is connected to an externally excited spring–pendulum system (three degree of freedom), subjected to harmonic excitation. The tuned absorber is usually designed to control one frequency at primary resonance where system damage is probable. Active control is also applied to the considered system via negative displacement feedback to change the linear frequency of the system and to shift it away from the resonating one. Also active control is applied to improve the behavior of the spring–pendulum at the primary resonance via negative velocity feedback or its square or cubic value. The multiple time scale perturbation technique is applied throughout. The stability of the system is investigated applying both frequency response function and phase-plane method. The effects of the absorber and different parameters on system behavior are studied numerically. Optimum working conditions of the system are extracted applying both passive and active control methods, to be used in the design of such systems.     

An inverted pendulum on a fixed and a moving base

The plane motions of a controlled single-link pendulum with a fixed suspension point and a pendulum with its suspension point located at the centre of a wheel which rolls without sliding along a flat horizontal surface are considered. The control torque, applied to the pendulum at the suspension point, is bounded in absolute magnitude. A controllability domain is constructed in the linear approximation for the one and the other pendulum, from all points of which the pendulum can be brought into the upper unstable equilibrium position without oscillations about the lower equilibrium. It is shown that the domain of controllability is greater for a pendulum mounted on a wheel, as a result it is more easily stabilizable. Control laws are constructed, under which the domain of attraction is identical to the controllability domain and is thereby the largest possible domain.     

Resonant oscillations of an extensible pendulum

Zusammenfassung Resonanzphänomene und Energieübertragung, assoziiert mit einem Paar gekoppelter nichtlinearer Differentialgleichungen, werden unter Anwendung asymptotischer Verfahren analysiert. Diese Gleichungen beschreiben die Bewegung des dehnbaren Pendels, das autoparametrische Erregung aufweist. Genaue Bedingungen werden gefunden, unter denen resonante oder nichtresonante Schwingungen entstehen. Diese Bedingungen hängen nicht nur von den physikalischen Parametern des Systems ab, sondern auch von der Energie der Schwingungen (das heisst von den Anfangsbedingungen). Im allgemeinen ist die Umhüllung der Resonanzschwingungen periodisch mit langen Perioden, und es besteht periodische Energieübertragung zwischen den zwei Freiheitsgraden.

---

---

Work sponsored by the U.S. Army Research Office (Durham).     

The nonsteady-state planar oscillations of a plate and oscillator on a two-layer base

We consider a plane dynamic contact problem for an inhomogeneous base of the following form: a soft layer on a rigid layer of an elastic half-plane. The layer is represented by a Winkler model, corresponding to the long-wave asymptotics of the equations of elasticity theory. The problem is reduced to a system of integro-differential equations that is solved numerically. We present the results of the computations of dynamic characteristics describing the oscillations of a rigid body and an oscillator on this base.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 64–68.     

The non-holonomic double pendulum: An example of non-linear non-holonomic system

An example of physically realizable non-linear non-holonomic mechanical system is proposed. The dynamical equations are written following a general method proposed in an earlier paper. In order to make this paper self-contained, an improved and shortened approach to the dynamics of non-holonomic systems is illustrated in preliminary sections.     

Three-dimensional non-linear oscillations of a rod with hinged supports

The free and forced flexural oscillations of a rod with hinged supports are investigated analytically and numerically. The geometrical non-linearity due to the change in the length of the central line of the rod accompanying its three-dimensional motion is taken into account. The oscillations of a rod with different natural frequencies in two mutually perpendicular directions as a consequence of the variance in the flexural stiffnesses of the rod or the stiffnesses of the supports in the different directions, are considered. It is shown in the case of natural oscillations that, together with two planar forms of motion, a form exists when a certain threshold value is exceeded, which corresponds to the motion of the cross-sections of the rod in a circle. The amplitude-frequency and phase-frequency characteristics of the system are constructed and qualitatively investigated in the neighbourhood of the principal resonance.     

On stability of motion of a symmetric body placed on a vibrating base

We consider the problem of stabilization of a symmetric solid body rotating about a fixed point and show that its unstable states can be stabilized by vertical vibration.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1661–1666, December, 1995.     

The stability of the parametric oscillations of second-order non-linear systems

The parametric oscillations of strongly non-linear systems with one degree of freedom are considered using a more general definition of these oscillations than the generally accepted definition. Stability criteria, that are verifiable using the signs of the derivatives of the amplitude-frequency characteristics, are found for the two families of periodic solutions corresponding to the fundamental parametric resonance. A condition is indicated under which the latter are monotonic and, as a result, one of the families is stable and the other is unstable. It is shown that, in a system with a concave non-monotonic elastic characteristic, the stable family loses stability for fairly large amplitudes and this effect is not revealed by the well-known analytical methods of non-linear mechanics.